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<widget class="widget-wrapper toc single" id="data-toc"><div class="widget-header cap dis-select"><span class="name">工程热力学前两章</span></div><div class="widget-body fs14"><div class="doc-tree active"><ol class="toc"><li class="toc-item toc-level-2"><a class="toc-link" href="#%E5%B8%B8%E7%94%A8%E8%AE%A1%E9%87%8F%E5%8D%95%E4%BD%8D"><span class="toc-text"> 常用计量单位</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#%E5%9F%BA%E6%9C%AC%E5%8D%95%E4%BD%8D"><span class="toc-text"> 基本单位</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E5%AF%BC%E5%87%BA%E5%8D%95%E4%BD%8D"><span class="toc-text"> 导出单位</span></a></li></ol></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E5%9F%BA%E6%9C%AC%E6%A6%82%E5%BF%B5%E4%BB%A5%E5%8F%8A%E5%AE%9A%E4%B9%89"><span class="toc-text"> 基本概念以及定义</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#%E7%83%AD%E5%8A%9B%E7%B3%BB%E7%BB%9F%E5%88%86%E7%B1%BB"><span class="toc-text"> 热力系统分类</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E7%83%AD%E5%8A%9B%E5%AD%A6%E7%8A%B6%E6%80%81"><span class="toc-text"> 热力学状态</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E5%9F%BA%E6%9C%AC%E7%8A%B6%E6%80%81%E5%8F%82%E6%95%B0"><span class="toc-text"> 基本状态参数</span></a><ol class="toc-child"><li class="toc-item toc-level-4"><a class="toc-link" href="#%E6%B8%A9%E5%BA%A6"><span class="toc-text"> 温度</span></a></li><li class="toc-item toc-level-4"><a class="toc-link" href="#%E5%8E%8B%E5%8A%9B"><span class="toc-text"> 压力</span></a></li><li class="toc-item toc-level-4"><a class="toc-link" href="#%E6%AF%94%E4%BD%93%E7%A7%AF"><span class="toc-text"> 比体积</span></a></li></ol></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E5%B9%B3%E8%A1%A1%E7%8A%B6%E6%80%81"><span class="toc-text"> 平衡状态</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E5%87%86%E5%B9%B3%E8%A1%A1%E8%BF%87%E7%A8%8B"><span class="toc-text"> 准平衡过程</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E5%8F%AF%E9%80%86%E8%BF%87%E7%A8%8B"><span class="toc-text"> 可逆过程</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E5%8A%9F"><span class="toc-text"> 功</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E7%83%AD%E9%87%8F"><span class="toc-text"> 热量</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E7%83%AD%E5%8A%9B%E5%BE%AA%E7%8E%AF"><span class="toc-text"> 热力循环</span></a></li></ol></li><li class="toc-item toc-level-2"><a class="toc-link" href="#%E7%83%AD%E5%8A%9B%E5%AD%A6%E7%AC%AC%E4%B8%80%E5%AE%9A%E5%BE%8B"><span class="toc-text"> 热力学第一定律</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#%E7%83%AD%E5%8A%9B%E5%AD%A6%E7%AC%AC%E4%B8%80%E5%AE%9A%E5%BE%8B%E8%A1%A8%E8%BE%BE%E5%BC%8F"><span class="toc-text"> 热力学第一定律表达式</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E7%83%AD%E5%8A%9B%E5%AD%A6%E5%8E%9F%E7%90%86%E5%88%86%E6%9E%90%E9%97%AE%E9%A2%98%E6%96%B9%E6%B3%95"><span class="toc-text"> 热力学原理分析问题方法</span></a><ol class="toc-child"><li class="toc-item toc-level-4"><a class="toc-link" href="#%E8%83%BD%E9%87%8F%E6%96%B9%E7%A8%8B%E7%9A%84%E7%AE%80%E5%8C%96"><span class="toc-text"> 能量方程的简化</span></a></li><li class="toc-item toc-level-4"><a class="toc-link" href="#%E7%83%AD%E5%8A%9B%E5%AD%A6%E7%AC%AC%E4%B8%80%E5%AE%9A%E5%BE%8B%E8%A1%A8%E8%BE%BE%E5%BC%8F%E5%92%8C%E9%80%82%E7%94%A8%E6%9D%A1%E4%BB%B6"><span class="toc-text"> 热力学第一定律表达式和适用条件</span></a></li><li class="toc-item toc-level-4"><a class="toc-link" href="#%E5%8F%AF%E9%80%86%E8%BF%87%E7%A8%8B%E4%B8%A4%E4%B8%AA%E7%83%AD%E5%8A%9B%E5%AD%A6%E5%BE%AE%E5%88%86%E5%85%B3%E7%B3%BB%E5%BC%8F"><span class="toc-text"> 可逆过程两个热力学微分关系式</span></a></li></ol></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E7%84%93"><span class="toc-text"> 焓</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E5%87%A0%E7%A7%8D%E5%8A%9F%E5%8F%8A%E5%85%B6%E5%85%B3%E7%B3%BB"><span class="toc-text"> 几种功及其关系</span></a></li></ol></li></ol></div></div></widget>




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<article class='md-text content post reveal'>
<h1 class="article-title"><span>工程热力学前两章</span></h1>
<h1 id="工程热力学基本概念和热力学第一定律"><a class="markdownIt-Anchor" href="#工程热力学基本概念和热力学第一定律"></a> 工程热力学基本概念和热力学第一定律</h1>
<p>写在前面:</p>
<iframe src="//player.bilibili.com/player.html?aid=768042394&bvid=BV1Rr4y1p7LW&cid=572574691&page=1" scrolling="no" border="0" frameborder="no" framespacing="0" allowfullscreen="true"> </iframe>
<p>参考视频系列如上</p>
<h1 id="第一章-热力学基本概念"><a class="markdownIt-Anchor" href="#第一章-热力学基本概念"></a> 第一章 热力学基本概念</h1>
<h2 id="常用计量单位"><a class="markdownIt-Anchor" href="#常用计量单位"></a> 常用计量单位</h2>
<h3 id="基本单位"><a class="markdownIt-Anchor" href="#基本单位"></a> 基本单位</h3>
<ul>
<li>长度 (m)</li>
<li>质量 (kg)</li>
<li>时间 (s)</li>
<li>热力学温度 (K)</li>
<li>物质的量 (mol)</li>
</ul>
<h3 id="导出单位"><a class="markdownIt-Anchor" href="#导出单位"></a> 导出单位</h3>
<ul>
<li>力=质量 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>×</mo></mrow><annotation encoding="application/x-tex">\times</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord">×</span></span></span></span> 加速度 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>1</mn><mi>N</mi><mo>=</mo><mn>1</mn><mi>k</mi><mi>g</mi><mo>⋅</mo><mi>m</mi><mi mathvariant="normal">/</mi><msup><mi>s</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">1 N =1 kg \cdot m/s^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord mathdefault" style="margin-right:0.10903em;">N</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord">1</span><span class="mord mathdefault" style="margin-right:0.03148em;">k</span><span class="mord mathdefault" style="margin-right:0.03588em;">g</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="mord mathdefault">m</span><span class="mord">/</span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></li>
<li>功=力 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>×</mo></mrow><annotation encoding="application/x-tex">\times</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord">×</span></span></span></span> 位移 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>1</mn><mi>J</mi><mo>=</mo><mn>1</mn><mi>k</mi><mi>g</mi><mo>⋅</mo><msup><mi>m</mi><mn>2</mn></msup><mi mathvariant="normal">/</mi><msup><mi>s</mi><mn>2</mn></msup></mrow><annotation encoding="application/x-tex">1J=1 kg \cdot m^2/s^2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord mathdefault" style="margin-right:0.09618em;">J</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord">1</span><span class="mord mathdefault" style="margin-right:0.03148em;">k</span><span class="mord mathdefault" style="margin-right:0.03588em;">g</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord">/</span><span class="mord"><span class="mord mathdefault">s</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></li>
<li>功率=功 / 时间 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mn>1</mn><mi>W</mi><mo>=</mo><mn>1</mn><mi>J</mi><mi mathvariant="normal">/</mi><mi>s</mi><mo>=</mo><mn>1</mn><mi>k</mi><mi>g</mi><mo>⋅</mo><msup><mi>m</mi><mn>2</mn></msup><mi mathvariant="normal">/</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">1W=1J/s=1 kg \cdot m^2/s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord">1</span><span class="mord mathdefault" style="margin-right:0.13889em;">W</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mord mathdefault" style="margin-right:0.09618em;">J</span><span class="mord">/</span><span class="mord mathdefault">s</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord">1</span><span class="mord mathdefault" style="margin-right:0.03148em;">k</span><span class="mord mathdefault" style="margin-right:0.03588em;">g</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span><span class="mord">/</span><span class="mord mathdefault">s</span></span></span></span></li>
</ul>
<h2 id="基本概念以及定义"><a class="markdownIt-Anchor" href="#基本概念以及定义"></a> 基本概念以及定义</h2>
<h3 id="热力系统分类"><a class="markdownIt-Anchor" href="#热力系统分类"></a> 热力系统分类</h3>
<p>按系统与外界之间的相互作用进行分类</p>
<ul>
<li>闭口系统: 质量没有越过边界<br />
也称为控制质量 (CM)系统</li>
<li>开口系统: 通过边界与外界有质量交换<br />
也称为控制容积 (CV)系统<br />
<em>判别标准为是否有质量越过边界</em><br />
<a target="_blank" rel="noopener" href="https://img.gejiba.com/image/EeC4sA"><img class="lazy" src="" data-src="https://img.gejiba.com/images/0c67c4cb5724ba5aaa24e25644af6c1a.md.png" alt="开口系与闭口系" /></a></li>
<li>绝热系统: 与外界无热量交换<br />
<em>判断标准为是否有热量越过边界</em><br />
<a target="_blank" rel="noopener" href="https://img.gejiba.com/image/EeCO13"><img class="lazy" src="" data-src="https://img.gejiba.com/images/0b492c111e87a23af988b8099e5e58fb.md.png" alt="绝热系统" /></a></li>
<li>孤立系统: 与外界无任何形式的质量和能量交换</li>
<li>简单可压缩系统: 由<mark>可压缩流体</mark>构成, 无化学反应, 与外界交换<mark>体积变化功</mark> (膨胀功或压缩功)的有限物质系统</li>
</ul>
<h3 id="热力学状态"><a class="markdownIt-Anchor" href="#热力学状态"></a> 热力学状态</h3>
<p>工质在热力变化过程中的<mark>某一瞬间所呈现的宏观物理状况</mark>, 简称状态</p>
<ol>
<li>状态参数: 描述工质所处<mark>状态的宏观物理量</mark></li>
<li>状态参数的特性:
<ul>
<li>物理上: 与过程无关</li>
<li>数学上: 积分与路径无关, 其微量是全微分</li>
</ul>
</li>
<li>常用的状态参数: 压力, 温度, 比体积, 热力学能, 焓, 熵</li>
<li><strong>压力, 温度, 比体积</strong>可以直接测量, 称为<mark>基本状态参数</mark></li>
</ol>
<h3 id="基本状态参数"><a class="markdownIt-Anchor" href="#基本状态参数"></a> 基本状态参数</h3>
<h4 id="温度"><a class="markdownIt-Anchor" href="#温度"></a> 温度</h4>
<ul>
<li>热力学定义: 处于<mark>同一热平衡</mark>状态的各个热力系, 必定有某一宏观特征彼此相同, 用于描述此宏观特征的物理量就是<mark>温度</mark></li>
<li>温标: 温度的数值表示法<br />
热平衡状态的判据, 即温度相等 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>⟷</mo></mrow><annotation encoding="application/x-tex">\longleftrightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.522em;vertical-align:-0.011em;"></span><span class="mrel">⟷</span></span></span></span> 热平衡</li>
<li><strong>热力学温标 (绝对温标)</strong>: 其确定的温度为热力学温度, 符号为 T, 单位是 K (开尔文), 依据热力学第二定律原理制定</li>
</ul>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>T</mi><mo>=</mo><mi>t</mi><mo>+</mo><mn>273.15</mn><mi>K</mi></mrow><annotation encoding="application/x-tex">T=t+273.15K
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.69841em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">t</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord">2</span><span class="mord">7</span><span class="mord">3</span><span class="mord">.</span><span class="mord">1</span><span class="mord">5</span><span class="mord mathdefault" style="margin-right:0.07153em;">K</span></span></span></span></span></p>
<h4 id="压力"><a class="markdownIt-Anchor" href="#压力"></a> 压力</h4>
<p><mark>绝对压力</mark> P, 大气压力 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>P</mi><mi>b</mi></msub></mrow><annotation encoding="application/x-tex">P_b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> ,表压力 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>P</mi><mi>e</mi></msub></mrow><annotation encoding="application/x-tex">P_e</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> (<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>P</mi><mo>&gt;</mo><msub><mi>P</mi><mi>b</mi></msub></mrow><annotation encoding="application/x-tex">P&gt;P_b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.72243em;vertical-align:-0.0391em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>) ,真空度 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>P</mi><mi>v</mi></msub></mrow><annotation encoding="application/x-tex">P_v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">v</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> (<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>P</mi><mo>&lt;</mo><msub><mi>P</mi><mi>b</mi></msub></mrow><annotation encoding="application/x-tex">P&lt;P_b</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.72243em;vertical-align:-0.0391em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>)<br />
<strong>只有绝对压力才是状态参数</strong>(包括 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>l</mi><mi>n</mi><mfrac><msub><mi>p</mi><mn>2</mn></msub><msub><mi>p</mi><mn>1</mn></msub></mfrac></mrow><annotation encoding="application/x-tex">ln \frac{p_2} {p_1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.228608em;vertical-align:-0.481108em;"></span><span class="mord mathdefault" style="margin-right:0.01968em;">l</span><span class="mord mathdefault">n</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7475em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.446108em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathdefault mtight">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.481108em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span> 中使用的为绝对压力)<br />
<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>P</mi><mo>=</mo><msub><mi>P</mi><mi>b</mi></msub><mo>+</mo><msub><mi>P</mi><mi>e</mi></msub></mrow><annotation encoding="application/x-tex">P = P_b + P_e</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">e</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 或 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>P</mi><mo>=</mo><msub><mi>P</mi><mi>b</mi></msub><mo>−</mo><msub><mi>P</mi><mi>v</mi></msub></mrow><annotation encoding="application/x-tex">P = P_b - P_v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">v</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></p>
<h4 id="比体积"><a class="markdownIt-Anchor" href="#比体积"></a> 比体积</h4>
<p>单位质量的工质所占有的体积, 符号是 v, 单位是 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msup><mi>m</mi><mn>3</mn></msup><mi mathvariant="normal">/</mi><mi>k</mi><mi>g</mi></mrow><annotation encoding="application/x-tex">m^3/kg</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.064108em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span></span></span></span></span><span class="mord">/</span><span class="mord mathdefault" style="margin-right:0.03148em;">k</span><span class="mord mathdefault" style="margin-right:0.03588em;">g</span></span></span></span></p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>v</mi><mo>=</mo><mfrac><mi>V</mi><mi>m</mi></mfrac><mo>=</mo><mfrac><mn>1</mn><mi>ρ</mi></mfrac></mrow><annotation encoding="application/x-tex">v = \frac{V} {m} = \frac{1} {\rho}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.04633em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">m</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.22222em;">V</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.20188em;vertical-align:-0.8804400000000001em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">ρ</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804400000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
<h3 id="平衡状态"><a class="markdownIt-Anchor" href="#平衡状态"></a> 平衡状态</h3>
<p><mark>没有外界作用</mark>(不包括重力场)条件下系统<mark>状态参数不随时间改变</mark>的状态</p>
<ul>
<li>热平衡: 无外界作用条件下, 系统内部, 系统与边界处处温度相等</li>
<li>力平衡: 无外界作用条件下, 系统内部, 系统与边界处处压力相等</li>
<li><strong>热力平衡的充要条件为系统同时达到热平衡和力平衡</strong><br />
温差——热不平衡势差<br />
压差——力不平衡势差</li>
</ul>
<p><mark>平衡状态的本质是不存在不平衡势差</mark><br />
<strong>实现平衡状态的充要条件为系统内部及系统与外界之间的一切不平衡势差消失</strong></p>
<blockquote>
<p>平衡状态和均匀状态的关系:<br />
1.平衡状态是相对<mark>时间</mark>而言的, 均匀状态是相对<mark>空间</mark>而言的<br />
2.对于气液<mark>两相</mark>并存的热力<mark>平衡</mark>系统, 气相和液相由于密度不同, 因此系统<mark>不是均匀</mark>的<br />
3. 对于处于热力<mark>平衡态</mark>的气体或液体 (单相), 如果不计算重力的影响, 则系统内部各处的性质是<mark>均匀</mark>一致的, 各处的温度, 压力, 比体积等状态参数都相同<br />
4. 本书在不加说明的情况下, 一律<strong>认为平衡状态下的单相物系统是均匀的</strong>, 各处的状态参数相同</p>
</blockquote>
<p><a target="_blank" rel="noopener" href="https://img.gejiba.com/image/EeCMsv"><img class="lazy" src="" data-src="https://img.gejiba.com/images/cdb04f408b2f9516ebac6a9f3dee6bbc.md.png" alt="平衡状态与均匀状态" /></a></p>
<blockquote>
<p>平衡状态与稳定状态的关系<br />
稳定: 状态参数不随时间变化<br />
平衡:<em>不存在不平衡势差是其本质</em>, 而状态参数不随时间变化只是现象<br />
<strong>稳定不一定平衡, 平衡一定稳定</strong></p>
</blockquote>
<p><a target="_blank" rel="noopener" href="https://img.gejiba.com/image/EeCfGh"><img class="lazy" src="" data-src="https://img.gejiba.com/images/465c914c8e605f34ebc78a07317a29f5.md.png" alt="465c914c8e605f34ebc78a07317a29f5.md.png" /></a><br />
对于和外界只有<mark>热量</mark>和<mark>体积变化功</mark>(膨胀功或压缩功)的<mark>简单可压缩系统</mark>, 只需要<mark>两个独立的参数</mark>(如 p,v; p, T 或 v,T)便可确定它的平衡状态 (例: 由已知 p,v 可求出 T,h,s,u)<br />
<strong>对于理想气体而言</strong> <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>p</mi><mi>v</mi><mo>=</mo><msub><mi>R</mi><mi>g</mi></msub><mi>T</mi></mrow><annotation encoding="application/x-tex">pv = R_gT</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">p</span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.15139200000000003em;"><span style="top:-2.5500000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.03588em;">g</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span></span></span></span>,实际气体需要已知 p,v 查表获取</p>
<ul>
<li>状态方程式: 对于简单可压缩系统, 表示<mark>状态参数之间关系</mark>的方程式</li>
</ul>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>p</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>v</mi><mo separator="true">,</mo><mi>T</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>T</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>p</mi><mo separator="true">,</mo><mi>v</mi><mo stretchy="false">)</mo><mo separator="true">,</mo><mi>F</mi><mo stretchy="false">(</mo><mi>p</mi><mo separator="true">,</mo><mi>v</mi><mo separator="true">,</mo><mi>T</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">p = f(v,T), T = f(p,v), F(p,v,T) = 0
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">p</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault">p</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord mathdefault">p</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span></p>
<ul>
<li>简单可压缩系统可以用平面坐标上一点确定其平衡状态 (实线表示可逆过程, 虚线表示不可逆过程)</li>
<li>在状态参数坐标图上, <strong>每一个点都代表一个平衡状态</strong><br />
<mark>非平衡状态没有确定状态参数, 不可表示在状态参数坐标图上</mark></li>
</ul>
<h3 id="准平衡过程"><a class="markdownIt-Anchor" href="#准平衡过程"></a> 准平衡过程</h3>
<ul>
<li>定义: 假定状态变化过程中, 所经历的每一个状态都<mark>无限接近平衡状态</mark>的过程</li>
<li>实现条件:<strong>推动过程进行的势差 (压差, 温差等)无限小</strong></li>
<li>有什么实际意义?
<ul>
<li>既是平衡, 又是变化</li>
<li>既可用状态参数描述, 又可进行热功转换</li>
</ul>
</li>
<li>准平衡过程在状态参数坐标图是可以近似用连续实线表示</li>
</ul>
<h3 id="可逆过程"><a class="markdownIt-Anchor" href="#可逆过程"></a> 可逆过程</h3>
<p>如果系统完成了某一过程后，<strong>可以沿原路逆行回复到原来的状态</strong>，并且<strong>不给外界留下任何变化</strong>，这样的过程为可逆过程<br />
<mark>可逆过程必然是准平衡过程，而准平衡过程不一定是可逆过程</mark><br />
准平衡过程 + 无耗散效应 = 可逆过程</p>
<h3 id="功"><a class="markdownIt-Anchor" href="#功"></a> 功</h3>
<ul>
<li>功的热力学定义：热力系统<mark>通过边界传递</mark>的能量，其全部效果可简单视为<mark>举起重物</mark>。符号 W，单位 J 或 kJ</li>
<li>功的符号：系统对外界做功为正，外界对系统作功为负</li>
<li><em>可逆</em>过程的<em>膨胀（压缩）功</em></li>
</ul>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>δ</mi><mi>w</mi><mo>=</mo><mfrac><mn>1</mn><mi>m</mi></mfrac><mi>p</mi><mi>d</mi><mi>V</mi><mo>=</mo><mi>p</mi><mi>d</mi><mi>v</mi><mo separator="true">;</mo><msub><mi>W</mi><mrow><mn>1</mn><mo>−</mo><mn>2</mn></mrow></msub><mo>=</mo><msubsup><mo>∫</mo><mn>1</mn><mn>2</mn></msubsup><mi>p</mi><mi>d</mi><mi>v</mi></mrow><annotation encoding="application/x-tex">\delta w = \frac{1} {m} p dV = p d v ; W_{1-2} = \int_{1}^{2} p dv
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.00744em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">m</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord mathdefault">p</span><span class="mord mathdefault">d</span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.902771em;vertical-align:-0.208331em;"></span><span class="mord mathdefault">p</span><span class="mord mathdefault">d</span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mbin mtight">−</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.4759580000000003em;vertical-align:-0.9119499999999999em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011249999999999316em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5640080000000003em;"><span style="top:-1.7880500000000001em;margin-left:-0.44445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span><span style="top:-3.8129000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9119499999999999em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">p</span><span class="mord mathdefault">d</span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span></span></span></span></span></p>
<ul>
<li>与 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>p</mi><mo>=</mo><mi>f</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p = f(v)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">p</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span></span></span></span> 有关, 是<mark>过程量</mark></li>
<li><mark>可逆功</mark>可用 p-v 图上过程线下面积表示<br />
<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>p</mi><mo>−</mo><mi>v</mi></mrow><annotation encoding="application/x-tex">p-v</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7777700000000001em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">p</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span></span></span></span> 图——示功图<br />
<a target="_blank" rel="noopener" href="https://img.gejiba.com/image/EeKpWC"><img class="lazy" src="" data-src="https://img.gejiba.com/images/ec0f9339e93dfd765392989bde94757f.png" alt="示功图" /></a></li>
</ul>
<h3 id="热量"><a class="markdownIt-Anchor" href="#热量"></a> 热量</h3>
<ul>
<li>定义: 热力系统与外界<strong>仅仅由于温差</strong>而<mark>通过边界传递</mark>的能量, 符号 Q, 单位 J 或 kJ</li>
<li>单位质量工质所传递的热量, 符号 q, 单位 J/kg 或 kJ/kg, 系统吸热 q&gt;0, 系统放热 q&lt;0</li>
<li><em>热量和功都是过程量而不是状态量</em></li>
<li><strong>可逆</strong>过程的热量: <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>δ</mi><mi>q</mi><mo>=</mo><mi>T</mi><mi>d</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">\delta q = T ds</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord mathdefault" style="margin-right:0.03588em;">q</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span><span class="mord mathdefault">d</span><span class="mord mathdefault">s</span></span></span></span>; <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>q</mi><mo>=</mo><msubsup><mo>∫</mo><mn>1</mn><mn>2</mn></msubsup><mi>T</mi><mi>d</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">q = \int_1^2 T ds</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">q</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.3648280000000002em;vertical-align:-0.35582em;"></span><span class="mop"><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0005599999999999772em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0090080000000001em;"><span style="top:-2.34418em;margin-left:-0.19445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.2579000000000002em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35582em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span><span class="mord mathdefault">d</span><span class="mord mathdefault">s</span></span></span></span></li>
<li><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">s</span></span></span></span>: 比熵, 定义式为 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>d</mi><mi>s</mi><mo>=</mo><mfrac><mrow><mi>δ</mi><mi>q</mi></mrow><mi>T</mi></mfrac></mrow><annotation encoding="application/x-tex">ds = \frac {\delta q} {T}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault">d</span><span class="mord mathdefault">s</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.277216em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9322159999999999em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.446108em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.03785em;">δ</span><span class="mord mathdefault mtight" style="margin-right:0.03588em;">q</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span>,单位 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>J</mi><mi mathvariant="normal">/</mi><mo stretchy="false">(</mo><mi>k</mi><mi>g</mi><mo>⋅</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">J/(kg \cdot K)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.09618em;">J</span><span class="mord">/</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03148em;">k</span><span class="mord mathdefault" style="margin-right:0.03588em;">g</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">K</span><span class="mclose">)</span></span></span></span> 或 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>k</mi><mi>J</mi><mi mathvariant="normal">/</mi><mo stretchy="false">(</mo><mi>k</mi><mi>g</mi><mo>⋅</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">kJ/(kg \cdot K)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.03148em;">k</span><span class="mord mathdefault" style="margin-right:0.09618em;">J</span><span class="mord">/</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.03148em;">k</span><span class="mord mathdefault" style="margin-right:0.03588em;">g</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">K</span><span class="mclose">)</span></span></span></span></li>
</ul>
<blockquote>
<p><strong>状态参数</strong>:<br />
<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>d</mi><mi>s</mi><mo>&gt;</mo><mn>0</mn><mo separator="true">,</mo><mi>δ</mi><mi>q</mi><mo>&gt;</mo><mn>0</mn><mo>→</mo></mrow><annotation encoding="application/x-tex">ds &gt; 0 ,\delta q &gt; 0 \rightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.73354em;vertical-align:-0.0391em;"></span><span class="mord mathdefault">d</span><span class="mord mathdefault">s</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord mathdefault" style="margin-right:0.03588em;">q</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">→</span></span></span></span> 系统吸热<br />
<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>d</mi><mi>s</mi><mo>=</mo><mn>0</mn><mo separator="true">,</mo><mi>δ</mi><mi>q</mi><mo>=</mo><mn>0</mn><mo>→</mo></mrow><annotation encoding="application/x-tex">ds = 0 ,\delta q = 0 \rightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault">d</span><span class="mord mathdefault">s</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord mathdefault" style="margin-right:0.03588em;">q</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">→</span></span></span></span> 系统绝热<br />
<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>d</mi><mi>s</mi><mo>&lt;</mo><mn>0</mn><mo separator="true">,</mo><mi>δ</mi><mi>q</mi><mo>&lt;</mo><mn>0</mn><mo>→</mo></mrow><annotation encoding="application/x-tex">ds &lt; 0 ,\delta q &lt; 0 \rightarrow</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.73354em;vertical-align:-0.0391em;"></span><span class="mord mathdefault">d</span><span class="mord mathdefault">s</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord">0</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord mathdefault" style="margin-right:0.03588em;">q</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">→</span></span></span></span> 系统放热<br />
<em>可逆绝热过程即为定熵过程</em></p>
</blockquote>
<ul>
<li><strong>可逆</strong>过程中单位质量工质与外界交换的热量可用 T-s 图上过程曲线下的面积表示</li>
</ul>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>δ</mi><mi>q</mi><mo>=</mo><mi>T</mi><mi>d</mi><mi>s</mi><mo separator="true">;</mo><mi>q</mi><mo>=</mo><msubsup><mo>∫</mo><mn>1</mn><mn>2</mn></msubsup><mi>T</mi><mi>d</mi><mi>s</mi></mrow><annotation encoding="application/x-tex">\delta q = T ds ; q = \int_1^2 T ds
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord mathdefault" style="margin-right:0.03588em;">q</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span><span class="mord mathdefault">d</span><span class="mord mathdefault">s</span><span class="mpunct">;</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">q</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.4759580000000003em;vertical-align:-0.9119499999999999em;"></span><span class="mop"><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011249999999999316em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5640080000000003em;"><span style="top:-1.7880500000000001em;margin-left:-0.44445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.8129000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.9119499999999999em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">T</span><span class="mord mathdefault">d</span><span class="mord mathdefault">s</span></span></span></span></span></p>
<p><a target="_blank" rel="noopener" href="https://img.gejiba.com/image/EeKLop"><img class="lazy" src="" data-src="https://img.gejiba.com/images/4f19ac8c025d0652321fd0ef7ac84de1.png" alt="T-s图" /></a></p>
<ul>
<li>T-s (温-熵) 图称为示热图</li>
</ul>
<h3 id="热力循环"><a class="markdownIt-Anchor" href="#热力循环"></a> 热力循环</h3>
<ul>
<li>定义: 工质由某一初态出发, 经过一系列热力状态变化后又<mark>回到原来初态</mark>的<mark>封闭</mark>热力过程</li>
<li><em>可逆循环与不可逆循环</em><br />
全部由可逆过程组成的循环为可逆循环, 循环中有任何一个过程不可逆都为不可逆循环. 可逆循环在状态参数图上为一封闭曲线</li>
</ul>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">经</mi><mi mathvariant="normal">济</mi><mi mathvariant="normal">性</mi><mi mathvariant="normal">指</mi><mi mathvariant="normal">标</mi><mi>η</mi><mo>=</mo><mfrac><mrow><mi mathvariant="normal">得</mi><mi mathvariant="normal">到</mi><mi mathvariant="normal">的</mi><mi mathvariant="normal">收</mi><mi mathvariant="normal">获</mi><mi>w</mi></mrow><mrow><mi mathvariant="normal">花</mi><mi mathvariant="normal">费</mi><mi mathvariant="normal">的</mi><mi mathvariant="normal">代</mi><mi mathvariant="normal">价</mi><mi>Q</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">经济性指标 \eta = \frac{得到的收获 w} {花费的代价 Q}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord cjk_fallback">经</span><span class="mord cjk_fallback">济</span><span class="mord cjk_fallback">性</span><span class="mord cjk_fallback">指</span><span class="mord cjk_fallback">标</span><span class="mord mathdefault" style="margin-right:0.03588em;">η</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.9880000000000002em;vertical-align:-0.8804400000000001em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.10756em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord cjk_fallback">花</span><span class="mord cjk_fallback">费</span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">代</span><span class="mord cjk_fallback">价</span><span class="mord mathdefault">Q</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord cjk_fallback">得</span><span class="mord cjk_fallback">到</span><span class="mord cjk_fallback">的</span><span class="mord cjk_fallback">收</span><span class="mord cjk_fallback">获</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8804400000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
<h1 id="第二章-热力学第一定律"><a class="markdownIt-Anchor" href="#第二章-热力学第一定律"></a> 第二章 热力学第一定律</h1>
<p>本质: 能量守恒与转换定律<br />
<mark>进入</mark>系统的能量-<mark>离开</mark>系统的能量=系统<mark>储存能量的变化</mark></p>
<h2 id="热力学第一定律"><a class="markdownIt-Anchor" href="#热力学第一定律"></a> 热力学第一定律</h2>
<h3 id="热力学第一定律表达式"><a class="markdownIt-Anchor" href="#热力学第一定律表达式"></a> 热力学第一定律表达式</h3>
<p>通用式如下, 其中 dE<sub>CV</sub>为系统能量增量<br />
<a target="_blank" rel="noopener" href="https://img.gejiba.com/image/EeKXEl"><img class="lazy" src="" data-src="https://img.gejiba.com/images/e736f06fbbfa1f66d1cceaad56d2a606.md.png" alt="热一律" /></a></p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>δ</mi><mi>Q</mi><mo>=</mo><mi>d</mi><msub><mi>E</mi><mrow><mi>C</mi><mi>V</mi></mrow></msub><mo>+</mo><msub><mrow><mo fence="true">(</mo><mi>h</mi><mo>+</mo><mfrac><msubsup><mi>c</mi><mi>f</mi><mn>2</mn></msubsup><mn>2</mn></mfrac><mo>+</mo><mi>g</mi><mi>z</mi><mo fence="true">)</mo></mrow><mrow><mi>o</mi><mi>u</mi><mi>t</mi></mrow></msub><mi>δ</mi><msub><mi>m</mi><mrow><mi>o</mi><mi>u</mi><mi>t</mi></mrow></msub><mo>−</mo><msub><mrow><mo fence="true">(</mo><mi>h</mi><mo>+</mo><mfrac><msubsup><mi>c</mi><mi>f</mi><mn>2</mn></msubsup><mn>2</mn></mfrac><mo>+</mo><mi>g</mi><mi>z</mi><mo fence="true">)</mo></mrow><mrow><mi>i</mi><mi>n</mi></mrow></msub><mi>δ</mi><msub><mi>m</mi><mrow><mi>i</mi><mi>n</mi></mrow></msub><mo>+</mo><mi>δ</mi><msub><mi>W</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\delta Q = dE_{CV} + \left(h+\frac{c_f^2} {2}+gz\right)_{out} \delta m_{out} - \left(h+\frac{c_f^2} {2}+gz\right)_{in} \delta m_{in} + \delta W_i
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord mathdefault">Q</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord mathdefault">d</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07153em;">C</span><span class="mord mathdefault mtight" style="margin-right:0.22222em;">V</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:3.0497300000000003em;vertical-align:-1.29973em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">(</span></span><span class="mord mathdefault">h</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6233239999999998em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.809216em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-2.4168920000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em;">f</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4192159999999999em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">g</span><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size4">)</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:-0.8691740000000001em;"><span style="top:-1.4002700000000001em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.29973em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:3.0497300000000003em;vertical-align:-1.29973em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">(</span></span><span class="mord mathdefault">h</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6233239999999998em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.809216em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-2.4168920000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em;">f</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4192159999999999em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">g</span><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size4">)</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:-0.8380660000000001em;"><span style="top:-1.4002700000000001em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.29973em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord"><span class="mord mathdefault">m</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></p>
<p>或</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">Φ</mi><mo>=</mo><mfrac><mrow><mi>δ</mi><mi>Q</mi></mrow><mrow><mi>d</mi><mi>τ</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>d</mi><msub><mi>E</mi><mrow><mi>C</mi><mi>V</mi></mrow></msub></mrow><mrow><mi>d</mi><mi>τ</mi></mrow></mfrac><mo>+</mo><msub><mrow><mo fence="true">(</mo><mi>h</mi><mo>+</mo><mfrac><msubsup><mi>c</mi><mi>f</mi><mn>2</mn></msubsup><mn>2</mn></mfrac><mo>+</mo><mi>g</mi><mi>z</mi><mo fence="true">)</mo></mrow><mrow><mi>o</mi><mi>u</mi><mi>t</mi></mrow></msub><msub><mi>q</mi><mrow><mi>m</mi><mo separator="true">,</mo><mi>o</mi><mi>u</mi><mi>t</mi></mrow></msub><mo>−</mo><msub><mrow><mo fence="true">(</mo><mi>h</mi><mo>+</mo><mfrac><msubsup><mi>c</mi><mi>f</mi><mn>2</mn></msubsup><mn>2</mn></mfrac><mo>+</mo><mi>g</mi><mi>z</mi><mo fence="true">)</mo></mrow><mrow><mi>i</mi><mi>n</mi></mrow></msub><msub><mi>q</mi><mrow><mi>m</mi><mo separator="true">,</mo><mi>i</mi><mi>n</mi></mrow></msub><mo>+</mo><mfrac><mrow><mi>δ</mi><msub><mi>W</mi><mi>i</mi></msub></mrow><mrow><mi>d</mi><mi>τ</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\Phi = \frac{\delta Q} {d \tau} = \frac{dE_{CV}} {d \tau} + \left(h+\frac{c_f^2} {2}+gz\right)_{out} q_{m,out} - \left(h+\frac{c_f^2} {2}+gz\right)_{in} q_{m,in} + \frac{\delta W_i} {d \tau}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord">Φ</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.0574399999999997em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault" style="margin-right:0.1132em;">τ</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord mathdefault">Q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.05744em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault" style="margin-right:0.1132em;">τ</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07153em;">C</span><span class="mord mathdefault mtight" style="margin-right:0.22222em;">V</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:3.0497300000000003em;vertical-align:-1.29973em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">(</span></span><span class="mord mathdefault">h</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6233239999999998em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.809216em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-2.4168920000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em;">f</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4192159999999999em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">g</span><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size4">)</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:-0.8691740000000001em;"><span style="top:-1.4002700000000001em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.29973em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.28055599999999997em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:3.0497300000000003em;vertical-align:-1.29973em;"></span><span class="minner"><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">(</span></span><span class="mord mathdefault">h</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6233239999999998em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.809216em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-2.4168920000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em;">f</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4192159999999999em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">g</span><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size4">)</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:-0.8380660000000001em;"><span style="top:-1.4002700000000001em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.29973em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:2.05744em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault" style="margin-right:0.1132em;">τ</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
<h3 id="热力学原理分析问题方法"><a class="markdownIt-Anchor" href="#热力学原理分析问题方法"></a> 热力学原理分析问题方法</h3>
<ol>
<li>首先选取适当热力系</li>
<li>写出相应的热一律通用能量方程</li>
<li>根据过程具体条件简化通用能量方程</li>
<li>通过推导演绎计算得到所讨论问题的结论</li>
</ol>
<h4 id="能量方程的简化"><a class="markdownIt-Anchor" href="#能量方程的简化"></a> 能量方程的简化</h4>
<p><strong>Case 1</strong>: <em>开口系</em>稳定流动<br />
开口系<mark>内部及其边界上</mark>各点工质的状态参数及运动参数都不随时间改变<br />
此时 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mfrac><mrow><mi>d</mi><msub><mi>E</mi><mi>c</mi></msub><mi>v</mi></mrow><mrow><mi>d</mi><mi>τ</mi></mrow></mfrac><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\frac{dE_cv} {d \tau} =0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.2412079999999999em;vertical-align:-0.345em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8962079999999999em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mathdefault mtight" style="margin-right:0.1132em;">τ</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.4101em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">d</span><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05764em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.16454285714285719em;"><span style="top:-2.357em;margin-left:-0.05764em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathdefault mtight">c</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span><span class="mord mathdefault mtight" style="margin-right:0.03588em;">v</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span> 且 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>q</mi><mrow><mi>m</mi><mo separator="true">,</mo><mi>i</mi><mi>n</mi></mrow></msub><mo>=</mo><msub><mi>q</mi><mrow><mi>m</mi><mo separator="true">,</mo><mi>o</mi><mi>u</mi><mi>t</mi></mrow></msub><mo>=</mo><msub><mi>q</mi><mi>m</mi></msub></mrow><annotation encoding="application/x-tex">q_{m,in} = q_{m,out} = q_m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.28055599999999997em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">m</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>,方程可简化为</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>q</mi><mo>=</mo><mi mathvariant="normal">Δ</mi><mi>h</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi mathvariant="normal">Δ</mi><msubsup><mi>c</mi><mi>f</mi><mn>2</mn></msubsup><mo>+</mo><mi>g</mi><mi mathvariant="normal">Δ</mi><mi>z</mi><mo>+</mo><msub><mi>w</mi><mi>i</mi></msub><mo>⟶</mo><mi>q</mi><mo>=</mo><mi mathvariant="normal">Δ</mi><mi>h</mi><mo>+</mo><msub><mi>w</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">q = \Delta h + \frac{1} {2} \Delta c_f^2 + g \Delta z + w_i \longrightarrow q = \Delta h + w_t
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">q</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em;"></span><span class="mord">Δ</span><span class="mord mathdefault">h</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:2.00744em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.864108em;"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em;">f</span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">g</span><span class="mord">Δ</span><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.661em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⟶</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">q</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em;"></span><span class="mord">Δ</span><span class="mord mathdefault">h</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></p>
<p>其中, <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>w</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">w_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 为对外做功, <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>w</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">w_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 为技术功<br />
<strong>Case 2</strong>:<em>闭口系</em>稳定流动<br />
闭口系下 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>q</mi><mrow><mi>m</mi><mo separator="true">,</mo><mi>i</mi><mi>n</mi></mrow></msub><mo>=</mo><msub><mi>q</mi><mrow><mi>m</mi><mo separator="true">,</mo><mi>o</mi><mi>u</mi><mi>t</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">q_{m,in} = q_{m,out} = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">i</span><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.28055599999999997em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">m</span><span class="mpunct mtight">,</span><span class="mord mathdefault mtight">o</span><span class="mord mathdefault mtight">u</span><span class="mord mathdefault mtight">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span>,通用式可简化为</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mfrac><mrow><mi>δ</mi><mi>Q</mi></mrow><mrow><mi>d</mi><mi>τ</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>d</mi><msub><mi>E</mi><mrow><mi>C</mi><mi>V</mi></mrow></msub></mrow><mrow><mi>d</mi><mi>τ</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mi>δ</mi><msub><mi>W</mi><mi>i</mi></msub></mrow><mrow><mi>d</mi><mi>τ</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\delta Q} {d \tau} = \frac{dE_{CV}} {d \tau} + \frac{\delta W_i} {d \tau}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0574399999999997em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault" style="margin-right:0.1132em;">τ</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord mathdefault">Q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.05744em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault" style="margin-right:0.1132em;">τ</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.07153em;">C</span><span class="mord mathdefault mtight" style="margin-right:0.22222em;">V</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:2.05744em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault" style="margin-right:0.1132em;">τ</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
<p>即 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>δ</mi><mi>Q</mi><mo>=</mo><mi>d</mi><mi>E</mi><mo>+</mo><mi>δ</mi><mi>W</mi><mo>→</mo><mi>q</mi><mo>=</mo><mi mathvariant="normal">Δ</mi><mi>h</mi><mo>+</mo><msub><mi>w</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">\delta Q = dE + \delta W \rightarrow q = \Delta h + w_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord mathdefault">Q</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">d</span><span class="mord mathdefault" style="margin-right:0.05764em;">E</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord mathdefault" style="margin-right:0.13889em;">W</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">q</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em;"></span><span class="mord">Δ</span><span class="mord mathdefault">h</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><br />
当忽略动位能变化时, 可化为 <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>δ</mi><mi>Q</mi><mo>=</mo><mi>d</mi><mi>U</mi><mo>+</mo><mi>δ</mi><mi>W</mi><mo>→</mo><mi>q</mi><mo>=</mo><mi mathvariant="normal">Δ</mi><mi>U</mi><mo>+</mo><mi>w</mi></mrow><annotation encoding="application/x-tex">\delta Q = dU + \delta W \rightarrow q = \Delta U + w</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord mathdefault">Q</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">d</span><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord mathdefault" style="margin-right:0.13889em;">W</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">q</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em;"></span><span class="mord">Δ</span><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span></span></span></span><br />
<mark>以上公式对于任何气体适用</mark><br />
<strong>Case 3</strong>: 循环系统<br />
循环系统中, <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>d</mi><msub><mi>E</mi><mrow><mi>c</mi><mi>v</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">dE_{cv} = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="mord mathdefault">d</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">c</span><span class="mord mathdefault mtight" style="margin-right:0.03588em;">v</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span>,in=out, 通用式可化简为</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>∮</mo><mfrac><mrow><mi>δ</mi><mi>Q</mi></mrow><mrow><mi>d</mi><mi>τ</mi></mrow></mfrac><mo>=</mo><mo>∮</mo><mfrac><mrow><mi>δ</mi><msub><mi>W</mi><mi>i</mi></msub></mrow><mrow><mi>d</mi><mi>τ</mi></mrow></mfrac><mo>⟶</mo><mo>∮</mo><mi>δ</mi><mi>Q</mi><mo>=</mo><mo>∮</mo><mi>δ</mi><mi>W</mi></mrow><annotation encoding="application/x-tex">\oint \frac{\delta Q} {d \tau} = \oint \frac{\delta W_i} {d \tau} \longrightarrow
\oint \delta Q = \oint \delta W
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.2336899999999997em;vertical-align:-0.86225em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011249999999999316em;">∮</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault" style="margin-right:0.1132em;">τ</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord mathdefault">Q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.23369em;vertical-align:-0.86225em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011249999999999316em;">∮</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault">d</span><span class="mord mathdefault" style="margin-right:0.1132em;">τ</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.13889em;">W</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⟶</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.22225em;vertical-align:-0.86225em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011249999999999316em;">∮</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord mathdefault">Q</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.22225em;vertical-align:-0.86225em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011249999999999316em;">∮</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord mathdefault" style="margin-right:0.13889em;">W</span></span></span></span></span></p>
<h4 id="热力学第一定律表达式和适用条件"><a class="markdownIt-Anchor" href="#热力学第一定律表达式和适用条件"></a> 热力学第一定律表达式和适用条件</h4>
<table>
<thead>
<tr>
<th style="text-align:center">表达式</th>
<th style="text-align:center">适用条件</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:center"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>q</mi><mo>=</mo><mi mathvariant="normal">Δ</mi><mi>u</mi><mo>+</mo><mi>w</mi></mrow><annotation encoding="application/x-tex">q = \Delta u + w</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">q</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em;"></span><span class="mord">Δ</span><span class="mord mathdefault">u</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span></span></span></span></td>
<td style="text-align:center">CM 系统, 1 kg 任何工质任何过程</td>
</tr>
<tr>
<td style="text-align:center"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>q</mi><mo>=</mo><mi mathvariant="normal">Δ</mi><mi>u</mi><mo>+</mo><msubsup><mo>∫</mo><mn>1</mn><mn>2</mn></msubsup><mi>p</mi><mi>d</mi><mi>v</mi></mrow><annotation encoding="application/x-tex">q = \Delta u + \int_1^2 p dv</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">q</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em;"></span><span class="mord">Δ</span><span class="mord mathdefault">u</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.3648280000000002em;vertical-align:-0.35582em;"></span><span class="mop"><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0005599999999999772em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0090080000000001em;"><span style="top:-2.34418em;margin-left:-0.19445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.2579000000000002em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35582em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">p</span><span class="mord mathdefault">d</span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span></span></span></span></td>
<td style="text-align:center">CM 系统, 1 kg 任何工质<mark>可逆</mark>过程</td>
</tr>
<tr>
<td style="text-align:center"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>q</mi><mo>=</mo><mi mathvariant="normal">Δ</mi><mi>h</mi><mo>+</mo><msub><mi>w</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">q = \Delta h + w_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">q</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em;"></span><span class="mord">Δ</span><span class="mord mathdefault">h</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></td>
<td style="text-align:center">1 kg 任何工质<mark>稳流</mark>过程</td>
</tr>
<tr>
<td style="text-align:center"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>q</mi><mo>=</mo><mi mathvariant="normal">Δ</mi><mi>u</mi><mo>+</mo><msub><mi>w</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">q = \Delta u + w_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">q</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em;"></span><span class="mord">Δ</span><span class="mord mathdefault">u</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></td>
<td style="text-align:center">1 kg 任何工质<mark>稳流</mark>过程, 忽略位动能变化</td>
</tr>
<tr>
<td style="text-align:center"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>q</mi><mo>=</mo><mi mathvariant="normal">Δ</mi><mi>h</mi><mo>−</mo><msubsup><mo>∫</mo><mn>1</mn><mn>2</mn></msubsup><mi>v</mi><mi>d</mi><mi>p</mi></mrow><annotation encoding="application/x-tex">q = \Delta h - \int_1^2 v dp</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">q</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.77777em;vertical-align:-0.08333em;"></span><span class="mord">Δ</span><span class="mord mathdefault">h</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.3648280000000002em;vertical-align:-0.35582em;"></span><span class="mop"><span class="mop op-symbol small-op" style="margin-right:0.19445em;position:relative;top:-0.0005599999999999772em;">∫</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0090080000000001em;"><span style="top:-2.34418em;margin-left:-0.19445em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.2579000000000002em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.35582em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mord mathdefault">d</span><span class="mord mathdefault">p</span></span></span></span></td>
<td style="text-align:center">1 kg 任何工质<mark>稳流可逆</mark>过程</td>
</tr>
</tbody>
</table>
<h4 id="可逆过程两个热力学微分关系式"><a class="markdownIt-Anchor" href="#可逆过程两个热力学微分关系式"></a> 可逆过程两个热力学微分关系式</h4>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo fence="true">{</mo><mtable rowspacing="0.24999999999999992em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>δ</mi><mi>q</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>d</mi><mi>u</mi><mo>+</mo><mi>p</mi><mi>d</mi><mi>v</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mi>δ</mi><mi>q</mi></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mi>d</mi><mi>h</mi><mo>−</mo><mi>v</mi><mi>d</mi><mi>p</mi></mrow></mstyle></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex">\left \{ \begin{aligned}
\delta q &amp; = du + p dv \\
\delta q &amp; = dh - v dp \\
\end{aligned}
\right .
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.00003em;vertical-align:-1.25003em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size4">{</span></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7500000000000002em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord mathdefault" style="margin-right:0.03588em;">q</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03785em;">δ</span><span class="mord mathdefault" style="margin-right:0.03588em;">q</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2500000000000002em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7500000000000002em;"><span style="top:-3.91em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathdefault">d</span><span class="mord mathdefault">u</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault">p</span><span class="mord mathdefault">d</span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathdefault">d</span><span class="mord mathdefault">h</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mord mathdefault">d</span><span class="mord mathdefault">p</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2500000000000002em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
<p>适合于<strong>闭口系统和稳流开口系统</strong></p>
<h3 id="焓"><a class="markdownIt-Anchor" href="#焓"></a> 焓</h3>
<ul>
<li>推动工质流动所做的功, 称为<mark>推动功</mark>, 推动功即为 pv</li>
<li>焓是物质进出开口系统时带入或带出的<em>热力学能与推动功之和</em>, 是<em>随物质一起转移的能量</em></li>
<li>焓是状态参数</li>
</ul>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>H</mi><mo>=</mo><mi>U</mi><mo>+</mo><mi>p</mi><mi>V</mi><mo>→</mo><mi>h</mi><mo>=</mo><mi>u</mi><mo>+</mo><mi>p</mi><mi>v</mi></mrow><annotation encoding="application/x-tex">H = U + p V \rightarrow h = u + p v
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.08125em;">H</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault" style="margin-right:0.10903em;">U</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">p</span><span class="mord mathdefault" style="margin-right:0.22222em;">V</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">→</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault">h</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault">u</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathdefault">p</span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span></span></span></span></span></p>
<h3 id="几种功及其关系"><a class="markdownIt-Anchor" href="#几种功及其关系"></a> 几种功及其关系</h3>
<table>
<thead>
<tr>
<th style="text-align:center">功</th>
<th style="text-align:center">表达式</th>
<th style="text-align:center">内容</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:center">膨胀功</td>
<td style="text-align:center"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>w</mi><mo>=</mo><msub><mi>w</mi><mi>t</mi></msub><mo>+</mo><mi mathvariant="normal">Δ</mi><mo stretchy="false">(</mo><mi>p</mi><mi>v</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">w = w_t + \Delta (p v)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Δ</span><span class="mopen">(</span><span class="mord mathdefault">p</span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span></span></span></span></td>
<td style="text-align:center">简单可压缩系统<strong>热变功的源泉</strong>,往往对应闭口系所求的功</td>
</tr>
<tr>
<td style="text-align:center">技术功</td>
<td style="text-align:center"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>w</mi><mi>t</mi></msub><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi mathvariant="normal">Δ</mi><msubsup><mi>c</mi><mi>f</mi><mn>2</mn></msubsup><mo>+</mo><mi>g</mi><mi mathvariant="normal">Δ</mi><mi>z</mi><mo>+</mo><msub><mi>w</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">w_t = \frac{1} {2} \Delta c_f^2 + g \Delta z + w_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2805559999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">t</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.2643239999999998em;vertical-align:-0.4192159999999999em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord">Δ</span><span class="mord"><span class="mord mathdefault">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8141079999999999em;"><span style="top:-2.4168920000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em;">f</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.4192159999999999em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">g</span><span class="mord">Δ</span><span class="mord mathdefault" style="margin-right:0.04398em;">z</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></td>
<td style="text-align:center">技术上可资利用的功</td>
</tr>
<tr>
<td style="text-align:center">流动功</td>
<td style="text-align:center"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>w</mi><mi>f</mi></msub><mo>=</mo><mi mathvariant="normal">Δ</mi><mo stretchy="false">(</mo><mi>p</mi><mi>v</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">w_f = \Delta (p v)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.10764em;">f</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Δ</span><span class="mopen">(</span><span class="mord mathdefault">p</span><span class="mord mathdefault" style="margin-right:0.03588em;">v</span><span class="mclose">)</span></span></span></span></td>
<td style="text-align:center">进出口推动功的差</td>
</tr>
<tr>
<td style="text-align:center">内部功</td>
<td style="text-align:center"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>w</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">w_i</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></td>
<td style="text-align:center">对机器内部所做的功; 忽略动位能变化时的技术功</td>
</tr>
<tr>
<td style="text-align:center">轴功</td>
<td style="text-align:center"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>w</mi><mi>s</mi></msub></mrow><annotation encoding="application/x-tex">w_s</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.02691em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">s</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></td>
<td style="text-align:center">通过轴与外界交换功; 内部功中有用部分; 开口系统所求的功</td>
</tr>
</tbody>
</table>



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